Definitions

Causality paradox

Causality (physics)

Causality describes the relationship between causes and effects, is fundamental to all natural science, especially physics, and has a basis in logic. It is also studied from the perspectives of philosophy, computer science, and statistics.

In physics causality is implemented by interpreting certain terms of a physical theory as causes and other terms as effects. Thus, in classical (Newtonian) mechanics a cause is represented by a force, an effect by the acceleration that can be derived from Newton's second law. For different physical theories the notions of cause and effect may be different. For instance, in Aristotelian physics the effect is not acceleration but velocity (one must push a cart twice as hard in order to have its velocity doubled). In the general theory of relativity, too, acceleration is not an effect (since it is not a generally relativistic vector); the general relativistic effects comparable to those of Newtonian mechanics are the deviations from geodesic motion in curved spacetime. As a consequence also uncaused motion is dependent on the theory: for Aristotle it is (absolute) rest, for Newton it is inertial motion (constant velocity with respect to an inertial frame of reference), in the general theory of relativity it is geodesic motion (to be compared with frictionless motion on the surface of a sphere at constant tangential velocity along a great circle).

A formulation of physical laws in terms of cause and effect is essential for the purposes of explanation and prediction. For instance, in Newtonian mechanics an observed acceleration can be explained by an applied force; Newton's second law can be used to predict the force necessary to realize a desired acceleration.

In classical physics a cause should always precede its effect, or at most be simultaneous with it (like force and acceleration in Newton's second law). In relativity theory this requirement is strengthened so as to restrict causes to be occurring in the backward (past) light cone of the event to be explained (effect); nor can an event be a cause of any event outside the former event's forward (future) light cone. These relativistic restrictions stem from the assumption that causal influences cannot travel faster than at the speed of light.

Another requirement, at least valid at the level of human experience, is that cause and effect be in close contact (requirement of contiguity). This requirement has been very influential in the past, in the first place as a result of direct observation of causal processes (like pushing a cart), in the second place as a problematic aspect of Newton's theory of gravitation (attraction of the earth by the sun by means of action at a distance) replacing mechanistic proposals like Descartes' vortex theory; in the third place as an incentive to develop dynamic field theories (e.g. Maxwell's electrodynamics and Einstein's general theory of relativity) restoring contiguity in the transmission of influences in a more successful way than did Descartes' theory.

As a consequence of an empiricist aversion of metaphysical explanations (like Descartes' vortex theory) the importance of causality has been downplayed (e.g. Newton's "Hypotheses non fingo"). According to Ernst Mach the notion of force in Newton's second law was pleonastic, tautological and superfluous. Indeed is it possible to consider the Newtonian equations of motion of the gravitational interaction between the sun (s) and a planet (p),

m_p frac{d^2 {mathbf r}_p }{ dt^2} = -frac{m_p m_s g ({mathbf r}_p - {mathbf r}_s)}{ |{mathbf r}_p - {mathbf r}_s|^3};; m_s frac{d^2 {mathbf r}_s }{dt^2} = -frac{m_s m_p g ({mathbf r}_s - {mathbf r}_p) }{ |{mathbf r}_s - {mathbf r}_p|^3},

as two coupled equations describing the positions scriptstyle {mathbf r}_p(t) and scriptstyle {mathbf r}_s(t) of planet and sun, without interpreting the right hand sides of these equations as forces; the equations just describe a process of interaction, without any necessity to interpret the sun as the cause of the motion of the planet (and vice versa), and allow to predict the states of the system s+p at later (as well as earlier) times.

The possibility of such a view is at the basis of the deductive-nomological (D-N) view of scientific explanation, considering an event to be explained if it can be subsumed under a scientific law. In the D-N view a physical state is considered to be explained if, applying the (deterministic) law, it can be derived from given initial conditions. Such `explanation by determinism' is sometimes referred to as causal determinism. A disadvantage of the D-N view is that causality and determinism are more or less identified. Thus, in classical physics, it was assumed that all events are caused by earlier ones according to the known laws of nature, culminating in Pierre-Simon Laplace's claim that if the current state of the world were known with precision, it could be computed for any time in the future or the past (see Laplace's demon). However, this is usually referred to as Laplace determinism (rather than `Laplace causality') because it hinges on determinism in mathematical models as dealt with in the mathematical Cauchy problem. Confusion of causality and determinism is particularly acute in quantum mechanics, this theory being acausal (as a consequence of its inability to provide descriptions of the causes of all actually observed effects) but deterministic in the mathematical sense.

In modern physics, the notion of causality had to be clarified. The insights of the theory of special relativity confirmed the assumption of causality, but they made the meaning of the word "simultaneous" observer-dependent. Consequently, the relativistic principle of causality says that the cause must precede its effect according to all inertial observers. This is equivalent to the statement that the cause and its effect are separated by a timelike interval, and the effect belongs to the future light cone of its cause. Equivalently, special relativity has shown that it is not only impossible to influence the past, it is also impossible to influence distant objects with signals that travel faster than the speed of light.

In the theory of general relativity, the concept of causality is generalized in the most straightforward way: the effect must belong to the future light cone of its cause, even if the spacetime is curved. New subtleties must be taken into account when we investigate causality in quantum mechanics and relativistic quantum field theory in particular. In quantum field theory, causality is closely related to the principle of locality. A careful analysis of the phenomena is needed, and the outcome slightly depends on the chosen interpretation of quantum mechanics: this is especially the case of the experiments involving quantum entanglement that require Bell's Theorem for their implications to be fully understood.

Despite these subtleties, causality remains an important and valid concept in physical theories. For example, the notion that events can be ordered into causes and effects is necessary to prevent causality paradoxes such as the grandfather paradox, which asks what happens if a time-traveller kills his own grandfather before he ever meets the time-traveller's grandmother. See also Chronology protection conjecture.

Distributed causality

Theories in physics like the Butterfly effect from chaos theory open up the possibility of a type of Distributed parameter systems in causality. The butterfly effect theory proposes:

"Small variations of the initial condition of a nonlinear dynamical system may produce large variations in the long term behavior of the system."
This opens up the opportunity to understand a distributed causality.

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